# Image processing - Why is sum of values of a blurring filter = 1?

Usually, blurring filters have the sum of all the values in the filter matrix equal to $$1$$.

Why is it so?

• The question contradicts the example, if the matrix that is provided is supposed to be a 3x3 convolution matrix. Can you please clarify? – A_A Oct 5 '18 at 12:25

## 2 Answers

Blurring an image means reducing its high frequencies while retaining its low frequencies.

Ususally this means a lowpass filter with a cutoff frequency of $$\omega_c$$. A standard low pass filter would have a DC response of $$1$$; i.e., $$H(0,0) = 1$$

This translates into time domain using the DTFT as: $$H(0,0) = 1 \implies \sum_n \sum_m h[n,m]e^{-j0n}e^{-j0n} =\sum_n \sum_m h[n,m] = 1$$

Which indicates that the sum of the impulse response samples equates to $$1$$.

For constant images to remain constant after blurring.

The most blurred images are flat-valued with any constant $$c$$. They are left invariant by traditional smoothers, low-pass or blur filters, hence $$\sum h_{k,l}c=c$$, for all $$c\neq 0$$ when $$\sum h_{k,l}=1$$,